Second Hardy–Littlewood conjecture

In number theory, the second Hardy–Littlewood conjecture is an important idea about how prime numbers are spread out. Primes are numbers greater than one that can only be divided evenly by one and themselves, like 2, 3, 5, and 7. The conjecture tries to answer questions about how many primes you can find in certain ranges or intervals of numbers.

This conjecture was suggested in 1923 by two famous mathematicians, G. H. Hardy and John Edensor Littlewood. They were studying patterns in prime numbers and wanted to understand how primes behave when you look at them closely over small ranges. Along with the first Hardy–Littlewood conjecture, this idea helps mathematicians predict where primes might appear.

Although we still don’t have a final proof for this conjecture, it remains an important puzzle in number theory. It helps guide research and understanding of the complex and fascinating world of prime numbers. Scientists and students continue to explore it, hoping to uncover more secrets about these building blocks of mathematics.

Statement

The second Hardy–Littlewood conjecture is an idea in number theory about prime numbers. It suggests a rule for how primes are spread out. Specifically, it says that for any whole numbers x and y that are at least 2, the number of primes up to x + y will be no more than the number of primes up to x added to the number of primes up to y. This is written using a special math symbol, but basically, it’s about counting primes in different ranges.

The idea was proposed by mathematicians G. H. Hardy and John Edensor Littlewood in 1923. It helps mathematicians understand how primes are distributed among larger numbers.

prime-counting function

Connection to the first Hardy–Littlewood conjecture

The second Hardy–Littlewood conjecture suggests that in looking at numbers, the primes (numbers only divisible by 1 and themselves) from a point like x + 1 up to x + y will never be more than the primes from 1 to y. This idea doesn’t match up with the first Hardy–Littlewood conjecture, which talks about finding groups of primes close together. We think the first real example where this happens would be with extremely large numbers.

For instance, there is a special set of 447 primes found within just 3159 numbers. If the first conjecture is true, we’d expect to see a similar set of primes when x is bigger than 1.5 × 10¹⁷⁴ but smaller than 2.2 × 10¹¹⁹⁸.

Generalization

The generalization of the second Hardy–Littlewood conjecture was proposed by S. I. Dimitrov in 2024. This idea suggests that for certain values, there is a large enough number that helps us understand how primes (numbers only divisible by 1 and themselves) are spread out in intervals.

The conjecture provides a mathematical relationship to predict the number of primes in combined intervals based on individual intervals, offering a new way to explore prime numbers.